Given a unital topological semigroup S and a unital semitopological semigroup T, whenever there is a morphism <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\rho : T\rightarrow \mathscr {S}urj(S)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>:</mo> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">S</mi> <mi>u</mi> <mi>r</mi> <mi>j</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the set of morphisms of S onto itself, we consider the semitopological semigroup <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S\rtimes _{\rho } T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mo>⋊</mo> <mi>ρ</mi> </msub> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> (the so called semidirect product of S and T with respect to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>) and give sufficient conditions that ensure the existence of a left invariant mean on the associated function spaces: <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$AP(S\rtimes _{\rho } T), WAP(S\rtimes _{\rho } T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <msub> <mo>⋊</mo> <mi>ρ</mi> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>W</mi> <mi>A</mi> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <msub> <mo>⋊</mo> <mi>ρ</mi> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$LUC(S\rtimes _{\rho } T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mi>U</mi> <mi>C</mi> <mo stretchy="false">(</mo> <mi>S</mi> <msub> <mo>⋊</mo> <mi>ρ</mi> </msub> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$WLUC(S\rtimes _{\rho } T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mi>L</mi> <mi>U</mi> <mi>C</mi> <mo stretchy="false">(</mo> <mi>S</mi> <msub> <mo>⋊</mo> <mi>ρ</mi> </msub> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>) of the semidirect product <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$S\rtimes _{\rho } T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mo>⋊</mo> <mi>ρ</mi> </msub> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation>.

On existence of invariant means on admissible function spaces of a semidirect product of topological semigroups

期刊论文
发表日期：

2026年5月18日

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Khadime Salame

摘要

Given a unital topological semigroup S and a unital semitopological semigroup T, whenever there is a morphism $\rho : T\rightarrow \mathscr {S}urj(S)$ $ρ : T \to S u r j (S)$ , the set of morphisms of S onto itself, we consider the semitopological semigroup $S\rtimes _{\rho } T$ $S ⋊_{ρ} T$ (the so called semidirect product of S and T with respect to $\rho $ $ρ$ ) and give sufficient conditions that ensure the existence of a left invariant mean on the associated function spaces: $AP(S\rtimes _{\rho } T), WAP(S\rtimes _{\rho } T)$ $A P (S ⋊_{ρ} T), W A P (S ⋊_{ρ} T)$ , $LUC(S\rtimes _{\rho } T)$ $L U C (S ⋊_{ρ} T)$ , and $WLUC(S\rtimes _{\rho } T$ $W L U C (S ⋊_{ρ} T$ ) of the semidirect product $S\rtimes _{\rho } T$ $S ⋊_{ρ} T$ .